To determine the key points on the graph of the quadratic function \( y = x^2 - 2x - 120 \), we need to find the vertex, x-intercepts, and y-intercept.
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Vertex: The vertex of a quadratic function in the form \( y = ax^2 + bx + c \) can be determined using the formula: \[ x = -\frac{b}{2a} \] In this case, \( a = 1 \) and \( b = -2 \): \[ x = -\frac{-2}{2 \cdot 1} = 1 \] To find the y-coordinate of the vertex, substitute \( x = 1 \) back into the equation: \[ y = 1^2 - 2 \cdot 1 - 120 = 1 - 2 - 120 = -121 \] Thus, the vertex is \( (1, -121) \).
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Y-Intercept: The y-intercept occurs when \( x = 0 \): \[ y = 0^2 - 2 \cdot 0 - 120 = -120 \] So, the y-intercept is \( (0, -120) \).
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X-Intercepts: To find the x-intercepts, set \( y = 0 \): \[ 0 = x^2 - 2x - 120 \] Factor the quadratic: \[ 0 = (x - 12)(x + 10) \] Setting each factor to zero gives: \[ x - 12 = 0 \quad \Rightarrow \quad x = 12 \] \[ x + 10 = 0 \quad \Rightarrow \quad x = -10 \] Therefore, the x-intercepts are \( (-10, 0) \) and \( (12, 0) \).
Summary of Key Points:
- Vertex: \( (1, -121) \)
- X-Intercepts: \( (-10, 0) \) and \( (12, 0) \)
- Y-Intercept: \( (0, -120) \)
In conclusion, the correct response includes the vertex as \( (1, -121) \), y-intercept as \( (0, -120) \), and x-intercepts as \( (-10, 0) \) and \( (12, 0) \).